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LAB 10 PRE-LAB

1. A spring is stretched to several different tensions, and its length is measured at each tension. The results are shown in the table. Plot the data as tension vs. length in the grid to the right. Scale your graph to use at least half of both axes. Draw a straight line to best fit the data.

Tension (N) Length (cm) 1.0 17.5 2.5 24.7

3.9 32.1 4.9 36.8

2. From the slope of the line, estimate the spring constant k of the spring.

k = N/m.

3. An oscillation has a frequency of 5 Hz. What is its period?

4. A wave has a frequency of 500 Hz and a wavelength of 0.60 m. What is the speed of the wave?

length (cm)

tens

ion

(N)

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LAB 10. VIBRATIONS AND WAVES

10.1 Problem

How can we measure the velocity of a wave?

How are the wavelength, period, and speed of a wave related?

What types of behavior do waves exhibit?

10.2 Equipment

spring with hook, several weights, meter stick, rod and table clamp; long coil spring mounted at one end to the wall; loose coil spring (Slinky), stopwatch; electronic frequency generator, oscillator, elastic cord, hanging weight for tension, table-mounted pulley; ripple tank, motorized straight pulse generator, straight barriers, plastic ruler

10.3 Apparatus

In this lab, you will use four systems. The first is a hanging spring with weights that can be suspended from it. The second, as shown in Figure 1, is a long coil spring that is attached to the wall at one end. You can create and examine a variety of waves in the spring.

The third system consists of a mechanical oscillator, frequency generator, stretchable string, clamp, pulley, and hanging weights, as shown schematically in Figure 2. This system will be used to generate standing waves on the string.

Figure 1. A coil spring.

pulley

frequency generator

vibrator

string

weight

Figure 2. Diagram of a mechanical oscillator and string.

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The fourth, as shown in Figure 3, is a ripple tank. The tank is a transparent pan of water which is elevated above the table. A motorized probe is used to disturb the surface of the water. A lamp is placed above and directed onto the tank, while a white screen is placed below the tank. The lamp and screen are used to project the water disturbance patterns in the same manner that a film is projected onto a movie screen. Several differently shaped barriers can be placed in the pan of water.

10.4 Background

Consider what happens when you toss a pebble into a still pond. The pebble disturbs the surface of the water, creating ripples. Picture the pattern of the ripples. Suppose a bug is floating on the water’s surface some distance away from the spot where you threw in the pebble. After the stone is tossed into the pond, the bug bobs up and down as the ripples pass the bug’s position. Why did the bug move up and down? How is this example different from the case of a bug that is pushed down a river by flowing water? A wave is a propagation of energy. Electromagnetic waves (light, radio, etc.) can propagate through vacuum; other types of waves need a medium to pass through. The wave is a disturbance in that medium. The ripples on the pond are an example of water waves.

Any wave shape that repeats itself is called periodic. The distance between successive crests, successive troughs, or any other pair of identical points on the wave is called the wavelength, λ. The maximum displacement of any point from the equilibrium position is called the wave amplitude, A.

The number of complete waves that pass a single position in a unit of time, such as a second, is the wave frequency, f. The time a single wave takes to pass that position is the wave period, T. The period is related to the frequency by T = 1/f.

Figure 3. A ripple tank.

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Waves may be either transverse, longitudinal, or a combination. In a transverse wave, the motion of individual points in the medium is perpendicular to the direction of propagation of the wave (i.e., up-down or left-right as the wave moves forward). In a longitudinal wave, the individual points move parallel to the direction of propagation (i.e., forward-backward as the wave moves forward). Instead of having crests and troughs, longitudinal waves have regions of compression and rarefaction. Many waves in nature, such as ocean waves, are a complex combination of these two limiting types.

10.5 Activities

You may do the following five stations in any order.

STATION 1: FORCE AND OSCILLATION OF A SPRING

External force applied to an object will change the object’s size or shape or both. Whether the object springs back to its original shape after the force is removed or remains deformed depends on the arrangement and bonding of the atoms in the material as well as the magnitude, rate and duration of force applied.

Force: 1. Record the initial length of the spring in centimeters in Table 1.

2. Hang a known mass from the spring and record the spring’s length. Add masses incrementally until you have four (4) measurements in addition to the starting length. Record these additional data in the table as well.

Table 1. Weight on a spring

measurement mass (g) length (cm) force (N)

initial 0

1

2

3

4

3. Convert the masses to the forces they exert on the spring by multiplying by the gravitational field g = 0.0098 N/g. In other words, F = mg. Enter these in the last column of Table 1.

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4. Plot a graph of cumulative load (force, not mass) vs. length. Enter the units in the axis labels. Scale the graph to use at least half of each axis. Title your graph.

5. Determine the slope of the plot. This is the spring constant k of your spring.

k =

Oscillation:

1. Hang a light mass on the same spring. When it comes to rest, the net force on it is zero: the downward force of gravity is exactly balanced by the upward force of the tension of the spring, and the weight does not accelerate. The position of the mass at which it experiences zero net force is its equilibrium position. The force of gravity actually does not complicate the system: the spring constant k of the spring is still the same. All that changes is the location of the equilibrium position.

load

(

)

length ( )

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2. Pull the mass downward gently and release. Find its period of oscillation. Do the same for three more masses, for a total of four (4) masses. Record the masses and periods in Table 2.

Table 2. Oscillating masses

Mass m (kg) Period T (s) T2 4π2m/k

3. Complete Table 2 by calculating T2 and 4π2m/k for each mass. Plot both quantities against mass. Draw a straight line through the 4π2m/k points. Do not connect the T2 points. They should lie close to the theoretical line. Check your calculations if they do not.

squa

re o

f per

iod

mass

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STATION 2: TRANSVERSE WAVES ON A SPRING 1. We begin by studying waves traveling in one dimension using the coil spring. Grasp the

free end of the coil with your hand and stand at a distance from the secured end so that the spring is slightly stretched. Generate single and multiple pulses with your hand. Try to make different wave shapes. What different shapes did you try to make? How did you try to make them? What shape were the resulting waves?

2. Estimate the speed of a single wave pulse by measuring how long it takes to travel a known distance. Show your measurements and calculations below.

3. Change the wave speed (not frequency: speed!) and describe what factors you must control to do this.

4. Place a piece of tape or yarn around one coil of the spring at any point along its length. Watch the motion of the tape as a wave pulse goes by. Does the tape move? If so, in what direction? Is this direction different from the direction that the wave travels?

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5. Generate another wave pulse on the spring. Observe the reflection of this pulse at the fixed end of the spring. The wave that arrives at the fixed end is called the incident wave. What is the displacement of the coils as the incident wave passes? Is it positive (up) or negative (down)? What is the direction of the displacement of the coils as the reflected wave passes? Describe both the incident and reflected waves. A sketch may help.

6. Generate a standing wave on the spring. The wave has a particular wavelength. What must you do to change the wavelength? Find two different ways to do this.

STATION 3: LONGITUDINAL WAVES ON A SPRING 1. Lay the Slinky on the floor and hold one end securely. Move the free end of the spring to

try to create a pulse that compresses the spring. How do you accomplish this? Is the coil compressed along its entire length at any instant?

2. Estimate the speed of one of the pulses. Do this by making rough measurements of the time the pulse takes to travel a known distance. Show your measurements, calculation, and estimate. Also determine if it is possible to change the wave speed, and describe what factors you must change to accomplish this.

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3. Tie a piece of yarn around one coil of the Slinky at any point along its length. Watch the tape as a wave pulse goes by. Does the tape move as the wave travels through it? If so, in what direction? Is this direction different from the direction that the wave pulse travels?

STATION 4: MEASURING WAVELENGTH AND FREQUENCY OF A STANDING WAVE

You will use the mechanical oscillator to move one end of an elastic string up and down. The string is held under tension by a weight on a pulley as shown in Figure 2. The frequency at which the oscillator oscillates is controlled electronically by the frequency generator. The wavelength of the standing waves can be measured by using a ruler or meter stick.

1. Make sure that the hanging weights do not touch the ground. Turn on the frequency generator. Experiment with frequencies ranging from a few hertz to a few hundred hertz. Do all frequencies create steady standing waves on the string?

2. Find a frequency at which a steady standing wave develops. Record this frequency in Table 1.

3. Measure the wavelength of the standing wave with a meter stick. Note that the distance between adjacent nodes (stationary positions) equals half a wavelength.

4. Change the frequency to create a different standing wave. Repeat parts 2 and 3 for the new standing wave.

5. Repeat parts 2 and 3 again with two more frequencies, for a total of four sets of data. Try to get a wide range of frequencies and wavelengths!

6. Calculate the period T (T = 1/frequency) of the wave for each set of measurements. Record the values in Table 1.

7. Calculate the speed (speed = distance/time = wavelength/period = wavelength · frequency) of the wave for each set of measurements. Record the values in Table 3.

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Table 3. Standing Waves in a string

Frequency (Hz) Wavelength (m) Period (s) Speed (m/s)

8. Make a graph of the wavelength (vertical axis) vs. period (horizontal axis) from your data in Table 3. Enter the units in the axis labels. Scale your graph to use at least half of each axis. Title your graph.

wav

elen

gth

(

)

period ( )

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9. What is the physical meaning of the slope of the graph?

10. If you change the period of a wave in the string, does the speed of the wave increase, decrease, or stay the same? Do your data and your graph support your answer?

STATION 5: RIPPLE TANK

1. Using the supplied motorized agitators, produce a series of straight wave pulses. Observe their propagation by looking at their projection on the white screen. Now produce a series of circular wave pulses by dipping a pen or other pointed object into the water. Sketch an instantaneous position of wave fronts for each type of pulse. Draw the wave rays on your sketch. Remember that wave rays show the direction the waves travel. Wave rays are always perpendicular to the wave fronts.

straight pulse circular pulse

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2. Using a single straight pulse, observe the incident and reflected waves as they collide with a straight barrier. Vary the angle between the straight barrier and the wave pulse. How is the direction of the reflected pulse affected? Sketch the wave pattern, showing both wave fronts (to show the momentary positions and shapes of the wave crests) and representative wave rays (to show the direction of wave travel).

3. Place two long, straight-edge barriers into the tank in a line parallel to the wave front of the straight pulse, leaving a small gap between the two barriers, as — —. Generate a single straight wave pulse. What happens to the pulse after it encounters the gap? Generate a train of straight wave pulses and observe it as it encounters the gap. Again, sketch the wave pattern, including wave fronts and representative wave rays.

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4. Using a single circular pulse, observe and describe the incident and reflected waves as they collide with a long, straight barrier. The reflected waves, like the incident waves, each make an arc of a circle. From what point do the reflected waves appear to emanate? (That is, where is the center of their circle?) Where does the center of the reflected waves appear to lie relative to the source (center) of the incident waves?

5. Sketch the pattern of the incident and reflected waves. Include in your diagram some representative rays for both the incident and reflected waves.